# FTOC, two parts

What is the difference between the two parts of FTOC, they seem to me to be essentially the same thing:

https://www.math.ucdavis.edu/~kouba/Math21BHWDIRECTORY/MVTFTC.pdf

No, they are not the same. FTC1 is the big gun: It states that if $f$ is continuous on $[a,b],$ then $f$ has an antiderivative $F,$ namely the function

$$F(x)= \int_a^x f(t)\, dt,\,\, x\in [a,b].$$

Recall that before the student has even seen FTC1, a lot of work has already been done in guaranteeing that the integrals $\int_a^x f(t)\, dt$ even exist (limits of Riemann sums and all that). FTC1 is a crowning acheivement in that it says not only do those integrals exist, the derivative of the function so formed gives us back $f.$

FTC2 is a lesser acheivement. All it says is that if you have any antiderivative $G$ of a continuous function $f$ on $[a,b],$ then $\int_a^b f(t)\,dt = G(b)-G(a).$ In FTC1 we already had an antiderivative, namely the $F$ defined there, which does this. FTC2 simply says any antiderivative will do this. The proof of FTC2 is almost trivial: By the MVT, any two antiderivatives on an interval differ by a constant, and the result follows.

If the function $f$ under consideration is continuous then the second part of FTOC is an immediate corollary of the first part. The real difference between both parts is visible when $f$ is discontinuous. I state these for Riemann integration.

First FTOC: If $f$ is Riemann integrable on $[a, b]$ and $$F(x) =\int_{a}^{x}f(t) \, dt$$ then $F$ is continuous on $[a, b]$ and if $f$ is continuous at $c\in[a, b]$ then $F$ is differentiable at $c$ and $F'(c) =f(c)$.

It should be noted that $F$ may be differentiable at some point $c$ even if $f$ is discontinuous at $c$, but then we can't say that $F'(c) = f(c)$ with guarantee. Further a subtle deep point: If a function is Riemann integrable then it is continuous at some points of the interval so that the equation $F'(c) =f(c)$ in the above statement of FTOC has relevance.

Second FTOC: If $F$ is differentiable on $[a, b]$ and the derivative $F'$ is Riemann integrable on $[a, b]$ then $$F(b) - F(a) = \int_{a}^{b}F'(x) \, dx$$

Note that $F'$ is not necessarily Riemann integrable and hence we have to assume its integrability in the statement of second FTOC above. Also note that even if $F'$ is assumed to be Riemann integrable it is not necessary that $F'$ is continuous on the whole interval $[a, b]$.

Both the parts of FTOC have independent proofs. It is only when the functions under integral sign are continuous that the second FTOC is a simple corollary of the first FTOC (you should verify that this is the case).

The statements and proofs of FTOC given your linked PDF deal with the simpler case when the functions being integrated are continuous. Your observation that they are almost same is correct. This is also evident from the proofs given in the link. The proofs of the theorems I have given above are somewhat difficult (compared to those given in pdf) and if you have a look at these proofs you will be at once convinced that both the parts of FTOC are reasonably different from each other.